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Casimir is a simple to install and use URL shortener, that let the user set the short URL he wants. It only needs PHP and MySQL to run.
xav.cc is an open-source URL shortening service, built on the PHP framework Symfony. The url shortener itself is available as the "sfDoctrineShortUrlPlugin" plugin for Symfony, and xav.cc adds some functionnalities : an api, an administration backend, etc.